On the genus and Hartshorne-Rao module of projective curves

Nadia Chiarli, Silvio Greco, Uwe Nagel

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23 Scopus citations


In this paper optimal upper bounds for the genus and the dimension of the graded components of the Hartshorne-Rao module of curves in projective n-space are established. This generalizes earlier work by Hartshorne [H] and Martin-Deschamps and Perrin [MDP]. Special emphasis is put on curves in P4. The first main result is a so-called Restriction Theorem. It says that a non-degenerate curve of degree d ≥ 4 in P4 over a field of characteristic zero has a non-degenerate general hyperplane section if and only if it does not contain a planar curve of degreed - 1 (see Th. 1.3). Then, using methods of Brodmann and Nagel, bounds for the genus and Hartshorne-Rao module of curves in Pn with non-degenerate general hyperplane section are derived. It is shown that these bounds are best possible in a very strict sense. Coupling these bounds with the Restriction Theorem gives the second main result for curves in P4. Then curves of maximal genus are investigated. The Betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in Pn of degree ≥ n + 2 is given. Finally, all pairs (d, g) of integers which really occur as the degree d and genus g of a non-degenerate curve in P4 are described.

Original languageEnglish
Pages (from-to)695-724
Number of pages30
JournalMathematische Zeitschrift
Issue number4
StatePublished - Dec 1998

ASJC Scopus subject areas

  • General Mathematics


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